Parallelogram平行四边形

QuadrilateralsAquadrilateral四边形isapolygonoffoursides.Bases底altitude高oppositeanglesParallelogram平行四边形Arhombus菱形isanequilateralparallelogramAsquare正方形isanequilateralandequiangularparallelogram.Also,itisanequilateralrectangle,oranequiangularrhombus.Thediagonalscutthecorneranglesinhalf,andtheycrossatrightangles.Akite筝形hastwopairsofadjacentsidesthatareequal.Akitehasonepairofequaloppositeangles.Thediagonalsofakitecrossatrightangles.Trapezoid梯形Anisoscelestrapezoid等腰梯形isatrapezoidhavingtwoequalnonparallelsides(legs腰).Thebaseanglesofanisoscelestrapezoidequal.Thediagonalsofanisoscelestrapezoidequal.Right-angledtrapezoid直角梯形ProvingquadrilateralstobeparallelogramsGeometryIdea8PropertiesofparallelogramBothpairsofoppositesidesareparallelOnepairofoppositesidesbothcongruentandparallelBothpairsofoppositesidesarecongruentBothpairsofoppositeanglescongruentDiagonalsbisecteachotherHowtoproveaquadrilateralisaparallelograms?AnotherpairofoppositesidescongruentAnotherpairofoppositesidesparallelAnotherpairofoppositeanglescongruentOnepairofoppositesidescongruent1Onepairofoppositesidesparallel1Onepairofoppositeanglescongruent?11EABDCEFABDCGAcounterexampleC'oCDABAnothercounterexampleDemonstratingtheneedtoconsiderallinformationgivenGeometryidea9MLPKCBAThreeequalcirclesA,B,andCmeetatacommonpointPandintersecteachother(pairwise)atpointsK,L,andM.ToproveABCKMLMLPKCBAQuadrilateralCLKAisaparallelogramCL//AK，CL=AKCL//PB,CL=PB,PB//AK,PB=AKBothquadrilateralsCLBP,APBKareparallelogramsMLPKCBAOParallelepiped平行六面体ThethreepointsofintersectionofthepairsofcircleslieonacircleequalinsizetotheotherthreecirclesWhereisthecenterofthefourthcircleMidlineofatrianglegeometryidea10MLJKGFIHquadrilateralmidpointsidediagonalsInsideparallelogramvertexMidlineKLisparalleltoFHandhashalfthelengthofFHInvariantsingeometryWhatareinvariantwhenwedistortthisfigureintoseveraldifferentlyshapedquadrilateralsABCD?Why?Underwhatconditionsthattheinnerquadrilateralisaarhombus,arectangle,asquare?EHGFPDCBAInnerquadrilateraloppositesidesLengthofthemedianofatrapezoidMedian:thesegmentthatjoinsthemidpointsofthenonparallelsidesofthetrapezoidMedianJLisparalleltothebasesThelengthofthemedianofatrapezoidistheaverageofthelengthsofthebasesLJFIHGGeometryIdea11NLJFIHGOPLJFIHGQRPLJFIHGPythagoreanTheorembcaDCABTheConverseofthePythagoreanTheoremForanythreepositivenumbersa,b,andcsuchthata2+b2=c2,thereexistsatrianglewithsidesa,bandc,andeverysuchtrianglehasarightanglebetweenthesidesoflengthsaandb.Inanyrighttriangle,theareaofthesquarewhosesideisthehypotenuse(thesideoppositetherightangle)isequaltothesumoftheareasofthesquareswhosesidesarethetwolegs(thetwosidesthatmeetatarightangle).Thesquareofthehypotenuseofarighttriangleisequaltothesumofthesquaresontheothertwosides.GeometryIdea12ProofusingsimilartrianglesbcaDCABaltituderightangle222abc2aBDc2bADcWedrawthealtitudefrompointC,andcallDitsintersectionwiththesideAB.ThenewtriangleACDissimilartoourtriangleABC,becausetheybothhavearightangle(bydefinitionofthealtitude),andtheysharetheangleatA,meaningthatthethirdanglewillbethesameinbothtrianglesaswell.Proofbyrearrangement赵爽勾股圆方图黄朱朱朱朱黄朱朱朱朱222214baabab刘徽对勾股定理的证明青朱青出青出朱出朱入青入青入thesumofequalsareequalBegintheclassbyaskingtheclassthequestion:WhatdoPythagoras,Euclid,andPresidentJamesA.Garfieldhaveincommon?What'stheanswer?AproofgivenbythePythagoreansbDCccaAbDCcLBBAGHEFGJFK2Sab22142baab2142cabAproofgivenbytheAmericanPresidentbaabccWecanproveitisatrapezoid22222221()()2111()2222()2SabababcababcababcAproofgivenbyEuclidsharethesamebaseandaltitudetriangle…mustbecongruenttotriangle…CongruentTrianglesTrianglesthathavethesamesizeandshape.Incongruenttriangles,all3pairsofcorrespondingangleshavethesamemeasure,andall3pairsofcorrespondingsideshavethesamemeasure.Side-Angle-Side(SAS)Iftwosidesandtheincludedangleofatrianglearecongruent,respectively,totwosidesandtheincludedangleofanothertriangle,thenthetwotrianglesarecongruent.Side-Side-Side(SSS)Ifthreesidesofatrianglearecongruent,respectively,tothethreesidesofanothertriangle,thenthetwotrianglesarecongruent.Angle-Angle-Side(AAS)Iftwoanglesandanon-includedsideofonetrianglearecongruenttotwoanglesandthecorrespondingnon-includedsideofasecondtriangle,thenthetwotrianglesarecongruent.Angle-Side-Angle(ASA)Iftwoanglesandtheincludedsideofatrianglearecongruenttotwoanglesandtheincludedsideofanothertriangle,thenthetwotrianglesarecongruent.Comparingareasofsimilarpolygons相似比theratioofsimilitude(theratioofanytwocorrespondinglinearparts)相似比的平方thesquareoftheratioofsimilitudebacDCAB222222222~~::::CDBADCACBACBCDBADCCDBADCACBSSSabcSSSkckakbcabGeometryIdea24Area1=Area2+Area3Area1:Area2:Area3=c2:a2:b2Area1=kc2,Area2=ka2,Area3=kb2c2=a2+b2Area1=Area2+Area3Area3Area2Area1bcaCABThefundamentaltermsPoint:apointhaspositiononly.Ithasnosize,nolength,width,orthickness.Line:alinehaslengthbuthasnowidthorthickness.Itmaybestraight,curved,oracombinationofthese.Itcouldbethoughtthatitisgeneratedbyamovingpoint.Straightline,curvedline,brokenlineOneandonlyone